Analogs of the Ikoma?Schwartz lemma and Liouville theorem for mappings with unbounded characteristic

Abstract

In the present paper, we obtain results on the local behavior of open discrete mappings $f:\;D \rightarrow \mathbb{R}^n, \quad n \geq 2,$, that satisfy certain conditions related to the distortion of capacities of condensers. It is shown that, in an infinitesimal neighborhood of zero, the indicated mapping cannot grow faster than an integral of a special type that corresponds to the distortion of the capacity under this mapping, which is an analog of the well-known growth estimate of Ikoma proved for quasiconformal mappings of the unit ball into itself and of the classical Schwartz lemma for analytic functions. For mappings of the indicated type, we also obtain an analogue of the well-known Liouville theorem for analytic functions.